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College Algebra Key Formulas and Concepts

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  • Definition of an even function
    A function is even if \(f(-x) = f(x)\) for all x in its domain.
  • Definition of an odd function
    A function is odd if \(f(-x) = -f(x)\) for all x in its domain.
  • Composite function notation and definition
    The composite function \((f \circ g)(x) = f(g(x))\) applies g first, then f.
  • Profit formula in terms of revenue and cost
    Profit is calculated as \(P(x) = R(x) - C(x)\), where R is revenue and C is cost.
  • Standard form equation of a circle
    The equation is \((x - h)^2 + (y - k)^2 = r^2\), where (h, k) is the center and r is the radius.
  • Formula for slope of a line
    Slope is \(m = \frac{y_2 - y_1}{x_2 - x_1}\) between two points \((x_1, y_1) and (x_2, y_2)\).
  • Slope-intercept form of a line
    The line is expressed as \(y = mx + b\), where m is slope and b is y-intercept.
  • Point-slope form of a line
    The line equation is \(y - y_1 = m(x - x_1)\), using slope m and point \((x_1, y_1)\).
  • Quadratic formula for roots
    Solutions to \(ax^2 + bx + c = 0\) are \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).
  • Axis of symmetry for a parabola
    The axis is the vertical line \(x = -\frac{b}{2a}\) for the quadratic \(ax^2 + bx + c\).
  • Vertex coordinates of a parabola
    Vertex is at \(\left(-\frac{b}{2a}, f\left(-\frac{b}{2a}\right)\right)\) for \(f(x) = ax^2 + bx + c\).
  • Difference quotient formula
    The difference quotient is \(\frac{f(x+h) - f(x)}{h}\), where \(h \neq 0\).
  • Average rate of change formula
    Average rate of change between a and b is \(\frac{f(b) - f(a)}{b - a}\), with \(a \neq b\).
  • Quadratic function in vertex form
    Expressed as \(f(x) = a(x - h)^2 + k\), where (h, k) is the vertex.
  • Present value formula with compound interest
    Present value is \(P = S \left(1 + \frac{r}{k}\right)^{-n}\), where n = kt and k is compounding periods per year.
  • Future value formula with compound interest
    Future value is \(S = P \left(1 + \frac{r}{k}\right)^{kt}\), with k compounding periods per year.
  • Future value with continuous compounding
    Future value is \(S = Pe^{rt}\), where e ≈ 2.718.
  • Logistic function formula
    The logistic function is \(f(x) = \frac{C}{1 + ae^{-bx}}\).
  • Properties of logarithms: log base b of b to x
    \(b^{\log_b x} = x\) and \(\log_b b^x = x\).
  • Natural logarithm notation
    The natural logarithm is \(\log_e x = \ln x\).
  • Logarithm of 1 to any base
    \(\log_b 1 = 0\) for any positive base b ≠ 1.
  • Exponential function general form
    An exponential function is \(f(x) = a b^x\), with \(b > 0, b \neq 1\).
  • Logarithmic function definition
    If \(y = \log_a x\), then \(x = a^y\).
  • Change-of-base formula for logarithms
    \(\log_b x = \frac{\log_a x}{\log_a b}\) for any positive bases a, b ≠ 1.
  • Logarithm of a product
    \(\log_b (MN) = \log_b M + \log_b N\).
  • Logarithm of a quotient
    \(\log_b \left(\frac{M}{N}\right) = \log_b M - \log_b N\).
  • Logarithm of a power
    \(\log_b M^k = k \log_b M\).